Roy Steiner equations for πn scattering

Transcription

1 Roy Steiner equations for N scattering C. Ditsche 1 M. Hoferichter 1 B. Kubis 1 U.-G. Meißner 1, 1 Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universität Bonn Institut für Kernphysik, Institute for Advanced Simulation, and Jülich Center for Hadron Physics, Forschungszentrum Jülich 7 th International Workshop on Chiral Dynamics 01 Jefferson Lab, August 8 th [JHEP 106 (01) 043] ( see also following talk by M. Hoferichter on [JHEP 106 (01) 063]) C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 1

2 Outline 1 Motivation: Why Roy Steiner equations for N scattering? Warm-up: Roy equations for scattering 3 N scattering basics 4 Roy Steiner equations for N scattering 5 Solving the t-channel Muskhelishvili Omnès problem 6 Summary & Outlook C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8

3 Motivation: Why N scattering? Why Roy Steiner equations? Renewed interest in N scattering: N N amplitudes e.g. for σ-term physics NN crossed amplitudes e.g. for nucleon form factors Need esp. low-energy (pseudophysical) amplitudes which are not very well known C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 3

4 Motivation: Why N scattering? Why Roy Steiner equations? Renewed interest in N scattering: N N amplitudes e.g. for σ-term physics NN crossed amplitudes e.g. for nucleon form factors Need esp. low-energy (pseudophysical) amplitudes which are not very well known Roy( Steiner) equations = Partial-Wave (Hyberbolic) Dispersion Relations coupled by unitarity and crossing symmetry PW(H)DRs together with unitarity, crossing symmetry, and chiral symmetry Can study processes at low energies with high precision: scattering: [Ananthanarayan et al. (001), García-Martín et al. (011)] K scattering: [Büttiker et al. (004)] γγ scattering: [Hoferichter et al. (011)] C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 3

5 Motivation: Why N scattering? Why Roy Steiner equations? Renewed interest in N scattering: N N amplitudes e.g. for σ-term physics NN crossed amplitudes e.g. for nucleon form factors Need esp. low-energy (pseudophysical) amplitudes which are not very well known Roy( Steiner) equations = Partial-Wave (Hyberbolic) Dispersion Relations coupled by unitarity and crossing symmetry PW(H)DRs together with unitarity, crossing symmetry, and chiral symmetry Can study processes at low energies with high precision: scattering: [Ananthanarayan et al. (001), García-Martín et al. (011)] K scattering: [Büttiker et al. (004)] γγ scattering: [Hoferichter et al. (011)] Roy Steiner equations for N scattering: Obtain low-energy (pseudophysical) amplitudes with better precision (update input & give errors) Framework allows for systematic improvements (subtractions, higher partial waves,...) C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 3

6 Warm-up: Roy equations for scattering (1) is fully crossing symmetric in Mandelstam variables s, t, and u = 4M s t Roy equations respect all available symmetry constraints: Lorentz invariance, unitarity, isospin & crossing symmetry, and (maximal) analyticity C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 4

7 Warm-up: Roy equations for scattering (1) is fully crossing symmetric in Mandelstam variables s, t, and u = 4M s t Roy equations respect all available symmetry constraints: Lorentz invariance, unitarity, isospin & crossing symmetry, and (maximal) analyticity Start from twice-subtracted fixed-t DRs of the generic form s + t + u = 4M = s + t + u T(s, t) = c(t) + 1 4M ds { s s s s + u s u } Im T(s, t) Determine subtraction functions c(t) via crossing symmetry PW expansion (I {0, 1, }, J = l): T I (s, t) = 3 ( ) (J + 1)P J cos θ(s, t) t I J (s) J=0 PW decomposition of these DRs yields the Roy equations [Roy (1971)] t I J (s) = ki J (s) + 1 I =0 J =0 4M ds K II JJ (s, s ) Im t I J (s ) Kernels: analytically known, contain Cauchy kernel KJJ II (s, s ) = δii δjj s +... s C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 4

8 Warm-up: Roy equations for scattering () t I J (s) = ki J (s) + 1 I =0 J =0 4M ds K II JJ (s, s ) Im t I J (s ) Validity: 4 M s 60 M (1.08 GeV) Mandelstam analyticity s 68 M (1.15 GeV) Subtraction constants (free parameters) contained in k I J (s): scattering lengths Matching to Chiral Perturbation Theory [Colangelo et al. (001)] C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 5

9 Warm-up: Roy equations for scattering () t I J (s) = ki J (s) + 1 I =0 J =0 4M ds K II JJ (s, s ) Im t I J (s ) Validity: 4 M s 60 M (1.08 GeV) Mandelstam analyticity s 68 M (1.15 GeV) Subtraction constants (free parameters) contained in k I J (s): scattering lengths Matching to Chiral Perturbation Theory [Colangelo et al. (001)] Elastic unitarity leads to coupled integral equations for the phase shifts δ I J (s) Im tj I (s) = σ (s) tj I (s) ( ) θ t 4M σ (s) tj I (s) = eiδi J (s) 1 = sin δj I i (s) eiδi J (s) t I J t I J σ (s) = 1 4M s C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 5

10 N scattering basics Generically: a (q) + N(p) b (q ) + N(p ) 00 Kinematics: s = (p + q), t = (p p ), u = (p q ) u = (m + M ) s t, Isospin structure: T ba = δ ba T + + iɛ bac τ c T Lorentz structure (I {+, }): { } T I = ū(p ) A I + /q +/q B I u(p) ν = s u 4m Crossing symmetry relates amplitudes for s-/u-channel (N N) and t-channel ( NN ), u M Π t crossing even and odd amplitudes: A ± (ν, t) = ±A ± ( ν, t), B ± (ν, t) = B ± ( ν, t) t u Ν s M Π s C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 6

11 N scattering basics: Subthreshold expansion 00 Subtraction of pseudovector Born terms: X X D ± = A ± + νb ± u Expand crossing even amplitudes {Ā+ } X I (ν, t), Ā ν, B + ν, B, D +, D ν around subthreshold point ν = t = 0: X I (ν, t) = x I mn (ν ) m t n m,n=0 u M Π t t Ν s M Π s C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 7

12 N scattering basics: Partial Waves PWs allow for easy incorporation of unitarity constraints helicity formalism [Jacob/Wick (1959)] C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 8

13 N scattering basics: Partial Waves PWs allow for easy incorporation of unitarity constraints helicity formalism [Jacob/Wick (1959)] s-channel PW projection: z s = cos θ s, W = s A I 1 l (s) = dz s P l (z s)a I (s, t) t=t(s,zs) 1 fl± {(E I (W) = 1 + m) [ A 16W I l (s) + (W m)bi l (s)] + (E m) [ A I l±1 (s) + (W + m)bi l±1 (s)]} MacDowell symmetry: fl+ I (W) = f (l+1) I ( W) l 0 [MacDowell (1959)] C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 8

14 N scattering basics: Partial Waves PWs allow for easy incorporation of unitarity constraints helicity formalism [Jacob/Wick (1959)] s-channel PW projection: z s = cos θ s, W = s A I 1 l (s) = dz s P l (z s)a I (s, t) t=t(s,zs) 1 fl± {(E I (W) = 1 + m) [ A 16W I l (s) + (W m)bi l (s)] + (E m) [ A I l±1 (s) + (W + m)bi l±1 (s)]} MacDowell symmetry: fl+ I (W) = f (l+1) I ( W) l 0 [MacDowell (1959)] t-channel PW expansion: z t = cos θ t A I (s, t) = 4 { } s=s(t,zt) p (J + 1)(p tq t) J P J (z t) f J + (t) m z t J J(J+1) tp J (z t) f J (t) B I (s, t) = 4 J+1 s=s(t,zt) (p J>0 J(J+1) tq t) J 1 P J (z t) f J (t) G-parity even J for I =+ (I t =0), odd J for I = (I t =1) C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 8

15 N scattering basics: Partial Waves PWs allow for easy incorporation of unitarity constraints helicity formalism [Jacob/Wick (1959)] s-channel PW projection: z s = cos θ s, W = s A I 1 l (s) = dz s P l (z s)a I (s, t) t=t(s,zs) 1 fl± {(E I (W) = 1 + m) [ A 16W I l (s) + (W m)bi l (s)] + (E m) [ A I l±1 (s) + (W + m)bi l±1 (s)]} MacDowell symmetry: fl+ I (W) = f (l+1) I ( W) l 0 [MacDowell (1959)] t-channel PW expansion: z t = cos θ t A I (s, t) = 4 { } s=s(t,zt) p (J + 1)(p tq t) J P J (z t) f J + (t) m z t J J(J+1) tp J (z t) f J (t) B I (s, t) = 4 J+1 s=s(t,zt) (p J>0 J(J+1) tq t) J 1 P J (z t) f J (t) G-parity even J for I =+ (I t =0), odd J for I = (I t =1) s-channel PW expansion and t-channel PW projection in analogy C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 8

16 Roy Steiner equations for N scattering: Hyperbolic DRs (Unsubtracted) Hyperbolic DRs: (s a)(u a) = b = (s a)(u a) with a, b R b = b(s, t, a) A + (s, t) = 1 [ 1 ds s s + 1 s u 1 ] s Im A + (s, t ) + 1 a (m+m ) 4M dt Im A+ (s, t ) t t B + (s, t) = N + (s, t) + 1 [ 1 ds s s 1 ] s Im B + (s, t ) + 1 dt ν Im B + (s, t ) u ν t t (m+m ) 4M [ ] N + (s, t) = g 1 and similarly for A, B, N [Hite/Steiner (1973)] m s 1 m u C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 9

17 Roy Steiner equations for N scattering: Hyperbolic DRs (Unsubtracted) Hyperbolic DRs: (s a)(u a) = b = (s a)(u a) with a, b R b = b(s, t, a) A + (s, t) = 1 [ 1 ds s s + 1 s u 1 ] s Im A + (s, t ) + 1 a (m+m ) 4M dt Im A+ (s, t ) t t B + (s, t) = N + (s, t) + 1 [ 1 ds s s 1 ] s Im B + (s, t ) + 1 dt ν Im B + (s, t ) u ν t t (m+m ) 4M [ ] N + (s, t) = g 1 and similarly for A, B, N [Hite/Steiner (1973)] Why HDRs? m s 1 m u Combine all physical regions important for reliable continuation to the subthreshold region [Stahov (1999)] Imaginary parts are only needed in regions where the corresponding PW decompositions converge Range of convergence can be maximized by tuning the free hyperbola parameter a Especially powerful for the determination of the σ-term [Koch (198)] C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 9

18 Roy Steiner equations for N scattering: Hyperbolic DRs (Unsubtracted) Hyperbolic DRs: (s a)(u a) = b = (s a)(u a) with a, b R b = b(s, t, a) A + (s, t) = 1 [ 1 ds s s + 1 s u 1 ] s Im A + (s, t ) + 1 a (m+m ) 4M dt Im A+ (s, t ) t t B + (s, t) = N + (s, t) + 1 [ 1 ds s s 1 ] s Im B + (s, t ) + 1 dt ν Im B + (s, t ) u ν t t (m+m ) 4M [ ] N + (s, t) = g 1 and similarly for A, B, N [Hite/Steiner (1973)] Why HDRs? m s 1 m u Combine all physical regions important for reliable continuation to the subthreshold region [Stahov (1999)] Imaginary parts are only needed in regions where the corresponding PW decompositions converge Range of convergence can be maximized by tuning the free hyperbola parameter a Especially powerful for the determination of the σ-term [Koch (198)] How to derive closed Roy Steiner system of PWHDRs: 1 Expand s-/t-channel imaginary parts of HDRs in s-/t-channel PWs, respectively Project nucleon pole terms and all imaginary parts onto both s- and t-channel PWs 3 Combine resulting RS equations with the s- & t-channel (extended) PW unitarity relations C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 9

19 Roy Steiner equations for N scattering: s-channel RS equations s-channel PW projection of pole terms and s-/t-channel-pw-expanded imaginary parts (unsubtracted) s-channel RS equations: f I l+ (W) = NI l+ (W) + 1 dt J { } G lj (W, t ) Im f J + (t ) + H lj (W, t ) Im f J (t ) + 1 m+m dw 4M l =0 { } Kll I (W, W ) Im fl I + (W ) + Kll I (W, W ) Im f(l I +1) (W ) = f(l+1) I ( W) l 0 [Hite/Steiner (1973)] Kernels: analytically known, e.g. K I ll (W, W ) = δ ll W W +... Validity: above threshold, assuming Mandelstam analyticity a = 3.19 M s [ (m + M ) = M, ] M W [ m + M = 1.08 GeV, 1.38 GeV ] C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 10

20 Roy Steiner equations for N scattering: t-channel RS equations t-channel PW projection of pole terms and s-/t-channel-pw-expanded imaginary parts (unsubtracted) t-channel RS equations: f J 0 + (t) = ÑJ + (t) + 1 dt } { K JJ 1 (t, t ) Im f J + (t ) + K JJ (t, t ) Im f J (t ) 4M J + 1 m+m dw { G Jl (t, W ) Im fl+ I (W ) + G Jl (t, W ) Im f(l+1) )} I (W l=0 f J 1 (t) = ÑJ (t) + 1 dt K 3 JJ (t, t ) Im f J (t ) 4M J >0 + 1 m+m dw { H Jl (t, W ) Im fl+ I (W ) + H Jl (t, W ) Im f(l+1) )} I (W l=0 Kernels analytically known, e.g. K 1 JJ (t, t ) = δ JJ t t +..., K 3 JJ (t, t ) = δ JJ t t +... Validity: above pseudothreshold, assuming Mandelstam analyticity a =.71 M t [ 4M, ] M t [ M = 0.8 GeV,.00 GeV ] C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 11

21 Roy Steiner equations for N scattering: Unitarity relations s-channel unitarity relations (I s {1/, 3/}): Im f Is l± (W) = qs f I s l± (W) ( ) θ W (m+m) fl± I fl± I [ ] 1 η Is l± (W) + θ ( W (m+m ) ) N 4q s N N +... C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 1

22 Roy Steiner equations for N scattering: Unitarity relations s-channel unitarity relations (I s {1/, 3/}): Im f Is l± (W) = qs f I s l± (W) ( ) θ W (m+m) fl± I fl± I [ ] 1 η Is l± (W) + θ ( W (m+m ) ) N 4q s N t-channel (extended) unitarity relations: (-body intermediate states: & KK +... ) N +... Im f J ± (t) = σ t ( t I J (t) ) f J ± (t) θ( t 4M ) + cj kt J σt K ( g I J (t) ) h J ± (t) θ( t 4M K) +... N N K f J ± t I J + h J ± g I J +... K N N Only linear in f J (t) less restrictive ± Watson s theorem: arg f J ± (t) = δi J (t) [Watson (1954)] for t < 16 M 40 M (0.88 GeV) C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 1

23 Roy Steiner equations for N scattering: Recoupling schemes s-channel subproblem: Kernels are diagonal for I {+, }, but unitarity relations are diagonal for I s {1/, 3/} All PWs are interrelated Once the t-channel PWs are known Structure similar to Roy equations f + 0+ f + 1+ f + 1 f 0+ f 1+ f 1 C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 13

24 Roy Steiner equations for N scattering: Recoupling schemes s-channel subproblem: Kernels are diagonal for I {+, }, but unitarity relations are diagonal for I s {1/, 3/} All PWs are interrelated Once the t-channel PWs are known Structure similar to Roy equations f + 0+ f + 1+ f + 1 f 0+ f 1+ f 1 t-channel subproblem: Only higher PWs couple to lower ones Only PWs with even or odd J are coupled No contribution from f J + to f J Leads to Muskhelishvili Omnès problem f 0 + f 1 + f + f 3 + f 1 f f 3 C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 13

25 Roy Steiner equations for N scattering: t-channel subproblem (1) Linear combinations Γ J (t) = m J J+1 f J (t) f J + (t) J 1 (unsubtracted) t-channel subproblem can be written as f+ 0 (t) = t 0 4m + (t) + dt Im f+ 0 (t ) (t 4m )(t t) Γ J 1 (t) = J t 4m Γ (t) + f J 1 (t) = J (t) + 1 4M 4M 4M dt Im Γ J (t ) (t 4m )(t t) dt Im f J (t ) t t [ ] f+ 0 (4m ) = 0 [ ] Γ J (4m ) = 0 with Im f J ± (t) = σ t ( t I J (t) ) f J ± (t) θ( t 4M ) +... Inhomogeneities (t): Born terms, s-channel integrals, and higher t-channel PWs; e.g. J (t) = ÑJ (t) + 1 dw { H Jl (t, W ) Im fl+ I (W ) + H Jl (t, W ) Im f )} (l+1) I (W m+m l=0 + 1 dt K 3 JJ (t, t ) Im f J (t ) 4M J J+ C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 14

26 Roy Steiner equations for N scattering: t-channel subproblem () In the low-energy (pseudophysical) region: Only the lowest s-/t-channel PWs are relevant Can match amplitudes to ChPT [Büttiker/Meißner (000), Becher/Leutwyler (001),...] Neglect inelasticities in both the - and the t-channel PWs ηj I (t) = 1 & no KK +... Watsons s theorem, single-channel approximation of t-channel subproblem C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 15

27 Roy Steiner equations for N scattering: t-channel subproblem () In the low-energy (pseudophysical) region: Only the lowest s-/t-channel PWs are relevant Can match amplitudes to ChPT [Büttiker/Meißner (000), Becher/Leutwyler (001),...] Neglect inelasticities in both the - and the t-channel PWs ηj I (t) = 1 & no KK +... Watsons s theorem, single-channel approximation of t-channel subproblem (Single-channel) Muskhelishvili Omnès problem with finite matching point t m [Muskhelishvili (1953), Omnès (1958), Büttiker et al. (004)] f(t) = (t)+ 1 t m 4M dt sin δ(t )e iδ(t ) f(t ) t + 1 t dt Im f(t ) t t t m f(t) e iδ(t) for t t m < t inel Solving for f(t) in 4M t tm requires: δ(t) for 4M t tm & Im f(t) for t tm Solution via once-subtracted Omnès function with t m < Ω(0) = 1 { t Ω(t) = exp t m 4M } { dt δ(t ) t t m t t = exp t 4M } dt δ(t ) t t e iδ(t) θ(t 4M )θ(tm t) t C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 15

28 Roy Steiner equations for N scattering: Subtractions In general: Subtractions May be necessary to ensure the convergence of DR/MO integrals asymptotic behavior Can be introduced to lessen the dependence of the low-energy solution on the high-energy behavior Parametrize high-energy information in (a priori unknown) subtraction constants matching to ChPT C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 16

29 Roy Steiner equations for N scattering: Subtractions In general: Subtractions May be necessary to ensure the convergence of DR/MO integrals asymptotic behavior Can be introduced to lessen the dependence of the low-energy solution on the high-energy behavior Parametrize high-energy information in (a priori unknown) subtraction constants matching to ChPT Favorable choice for t-channel MO problem: subthreshold expansion around ν = t = 0 Subtract HDRs for A ± and B ± at s = u = m + M and t = 0 Done up to full second order; added (partial) third subtraction for A ± Obtain sum rules for subthreshold parameters x I mn General structure of RS/MO problem remains unchanged HDRs s-/t-channel RS equations (pole terms & kernels) t-channel MO problem, e.g. for P-waves (n 1): Γ 1 (t) = 1 Γ (t) n-sub + tn 1 (t 4m ) dt Im Γ 1 (t ) t n 1 (t 4m )(t t) f 1 (t) = 1 (t) n-sub + tn 4M 4M dt Im f 1 (t ) t n (t t) C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 16

30 Roy Steiner equations for N scattering: Solution strategy scattering phases δ It J (t) higher partial waves Imf J>Jd ± (t t m ) t-channel partial waves f J ± (t) high-energy region 1. solve RS (MO) equations for J J d and t t m Imf±(t J t m ) inelasticities for t t m and s s m Imf 3. N coupling and subthreshold parameters higher partial waves Imf I (l>ld)± (s s m) s-channel partial waves f I l± (s) high-energy region. solve RS equations for l l d and s s m Imfl± I (s s m) C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 17

31 t-channel Muskhelishvili Omnès problem: Input Here, show results for the P-waves, since S-wave: Strong effect from KK intermediate states (f 0(980) resonance) need two-channel MO analysis following talk P-waves: Single-channel MO approximation well justified in the low-energy region D-waves: Dominated by nucleon pole terms in general for all PWs for t 4M First step: Check consistency with KH80 t-channel PWs iteration with s-channel results t.b.d. C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 18

32 t-channel Muskhelishvili Omnès problem: Input Here, show results for the P-waves, since S-wave: Strong effect from KK intermediate states (f 0(980) resonance) need two-channel MO analysis following talk P-waves: Single-channel MO approximation well justified in the low-energy region D-waves: Dominated by nucleon pole terms in general for all PWs for t 4M First step: Check consistency with KH80 t-channel PWs iteration with s-channel results t.b.d. Input used: phase shifts δ I J [Caprini/Colangelo/Leutwyler (in preparation)] s-channel: SAID PWs [Arndt et al. (008)] for W.5 GeV, above: Regge model [Huang et al. (010)] KH80 [Höhler (1983)] subthreshold parameters & coupling g /(4) = 14.8 modern value: g /(4) = 13.7 ± 0. [Baru et al. (011)] t-channel: All contributions above t m = 0.98 GeV set to zero solutions fixed f I J (tm) = 0 C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 18

33 t-channel Muskhelishvili Omnès problem: P-waves f J + less well determined in MO framework than f J, since Effectively one subtraction less introduced partial third subtraction Enhanced sensitivity to subtraction constants Ñ 0 + (4M ) = ÑJ Γ (4M ) = 0 Estimate systematic uncertainties (1): fixed-t limit a modulo t-channel integrals Estimate systematic uncertainties (): Variation of the matching point t m similar... MO solutions in general consistent with KH80 results C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 19

34 t-channel Muskhelishvili Omnès problem: Isovector spectral functions P-waves feature in dispersive analyses of the Sachs form factors of the nucleon: Im G v E (t) = q3 t m ( F V (t) ) f 1 t + (t) θ ( t 4M) Im G v M (t) = q3 t ( F V (t) ) f 1 (t) θ ( t 4M ) t C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 0

35 Summary & Outlook What has been done: Derived a closed system of Roy Steiner equations (PWHDRS) for N scattering Constructed unitarity relations including KK intermediate states for the t-channel PWs Optimized the range of convergence by tuning a for s- and t-channel each Implemented subtractions at several orders Solved the t-channel (single-channel) MO problem t-channel RS/MO machinery works modulo the S-wave C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 1

36 Summary & Outlook What has been done: Derived a closed system of Roy Steiner equations (PWHDRS) for N scattering Constructed unitarity relations including KK intermediate states for the t-channel PWs Optimized the range of convergence by tuning a for s- and t-channel each Implemented subtractions at several orders Solved the t-channel (single-channel) MO problem t-channel RS/MO machinery works modulo the S-wave What needs to be done: Two-channel MO analysis for the S-wave, effect on scalar form factor following talk Numerical solution of the s-channel subproblem using the t-channel results as input Self-consistent, iterative solution of the full RS system lowest PWs & low-energy parameters Possible improvements: Higher subtractions, higher PWs, more inelastic input,... C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 1

37 N scattering basics Generically: a (q) + N(p) b (q ) + N(p ) 00 Kinematics: s = (p + q), t = (p p ), u = (p q ) u = (m + M ) s t, Isospin structure: T ba = δ ba T + + iɛ bac τ c T Lorentz structure (I {+, }): { } T I = ū(p ) A I + /q +/q B I u(p) ν = s u 4m Crossing symmetry relates amplitudes for s-/u-channel (N N) and t-channel ( NN ), u M Π t u t crossing even and odd amplitudes: A ± (ν, t) = ±A ± ( ν, t), B ± (ν, t) = B ± ( ν, t) Isospin & crossing Ν s M Π AI=+ A I= = 1 1 AIs=1/ A Is=3/ = 6 AIt=0 0 1 A It=1, A {A, B} s C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8

38 N scattering basics: Subthreshold expansion Subtraction of pseudovector Born terms: X X 00 D ± = A ± + νb ± Expand crossing even amplitudes {Ā+ X I (ν, t), Ā ν, B + ν, B, D +, D ν around subthreshold point ν = t = 0: X I (ν, t) = xmn I (ν ) m t n m,n=0 Relations between subthreshold parameters x I mn : d + mn = a + mn + b + m 1,n d + 0n = a+ 0n d mn = a mn + b mn Subthreshold expansion of A ± and B ± : } u M Π t u t A + (ν, t) = g m + d d+ 01 t + a+ 10 ν + O ( ν 4, ν t, t ) A (ν, t) = a 00 ν + a 01 νt + O( ν 3, νt ) B + (ν, t) = g 4mν M M + b + 00 ν + O( ν 3, νt ) [ B (ν, t) = g m g M + t ] M + b 00 + b 01 t + O( ν, ν t, t ) Ν s M Π s C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 3

39 Roy Steiner equations for N scattering: Range of convergence Assumption: Mandelstam analyticity [Mandelstam (1958,1959)] T(s, t) = 1 ds du ρ su(s, u ) (s s)(u u) + 1 dt du ρ tu(t, u ) (t t)(u u) + 1 ds dt ρ st(s, t ) (s s)(t t) with integration ranges defined by the support of the double spectral regions ρ Boundaries of ρ are given by the lowest graphs (I) (II) (III) (IV) 100 Ρut Ρsu Convergence of PW exps. of imaginary parts Lehman ellipses for z = cos θ [Lehmann (1958)] Convergence of PW projs. of full equations u M Π 50 0 t Ν Ρst for given a, hyperbolas must not enter any ρ 50 for all needed values of b Constraints on b yield ranges in s & t s M Π C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 4

40 t-channel Muskhelishvili Omnès problem: P-waves () Estimate systematic uncertainties: Variation of the matching point t m effect of f I J (t m) = 0 Convergence pattern & internal consistency Consistency with KH80 MO solutions in general consistent with KH80 results C. Ditsche (HISKP & BCTP, Uni Bonn) Roy Steiner equations for N scattering Chiral Dynamics 01, JLab, August 8 5